A Frobenius-Type Formula for Compact Lie Groups

Abstract

Let $G$ be a group and $\hat{I}\pm :G\times G\to G$ denote the commutator map. In the case of finite groups, Frobenius gave the formula to compute the cardinalities of the fibres $\hat{I}{\pm }^{-1}(g)$ in terms of the character values $\ddot{I}‡(g)$ for irreducible characters $\ddot{I}‡$ of $G$. We generalise this formula to compact Lie groups. Further, we connect this generalised formula to the commutator probability of the concerned groups.

14 Pages